Timeline for Existence of connections on principal bundles
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Mar 31, 2011 at 16:31 | answer | added | bavajee | timeline score: 8 | |
| Mar 17, 2011 at 16:45 | comment | added | agt | @BS:Aren't such objects known as Ehresmann connections? | |
| Mar 17, 2011 at 16:31 | comment | added | BS. | I think the picture is clear if you view a connection as a projection from the tangent bundle $TE$ to its vertical subbundle (for principal bundles one adds in general an invariance condition under the group). If your $\omega_\alpha$ are the projections given by local trivializations (or the closely related Lie-agebra valued 1-forms), the "right hand side" is still a projection, since the $f_\alpha$ sum to $1$. | |
| Mar 17, 2011 at 9:34 | comment | added | agt | The first step in your proof should be corrected in: Let us choose a atlas of trivializing chart $\{(U_\alpha,\phi_\alpha)\}$ for the fiber bundle. | |
| Mar 17, 2011 at 9:26 | answer | added | agt | timeline score: 0 | |
| Mar 16, 2011 at 14:27 | answer | added | Florian | timeline score: 0 | |
| Mar 16, 2011 at 0:24 | comment | added | José Figueroa-O'Farrill | Section 2 of Chapter II in Volume 1 of Kobayashi Nomizu's Foundations of Differential Geometry proves that connections exist on principal fibre bundles over a paracompact base. | |
| Mar 15, 2011 at 22:55 | comment | added | Kevin Wray | Yes, I meant right-hand side (it's corrected now). To me the right-hand side looks very weird. First, shouldn't $f_\alpha$ and $\pi$ be the corresponding maps on the diff. forms? Also, if you can define $(f_\alpha \circ \pi)\omega_\alpha$, then wouldn't this object live on some open set of $M$? | |
| Mar 15, 2011 at 22:50 | history | edited | Kevin Wray | CC BY-SA 2.5 |
added 1 characters in body; added 1 characters in body
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| Mar 15, 2011 at 22:50 | comment | added | Dan Ramras | Did you mean to write "right-hand side"? Since $\omega_\alpha$ is a (local) connection on the bundle, when you sum over the partition of unity you should still be getting an object on the bundle. To be precise, though, you have to think about what exactly $\omega_\alpha$ means. It should be a rule for lifting tangent vectors in $M$ to tangent vectors in $E$. If you write it out carefully, you should see that your idea works. Good sources for connections on principal bundles are Spivak, volume 2 (chapter 9 I think) and Atiyah and Bott's paper on Yang-Mills theory (section 3, I think). | |
| Mar 15, 2011 at 22:39 | history | asked | Kevin Wray | CC BY-SA 2.5 |